Degrees: Measuring Angles
We measure the size of an angle using degrees.
Example: Here are some examples of angles and their degree measurements.
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Acute Angles
An acute angle is an angle measuring between 0 and 90 degrees.
Example:
The following angles are all acute angles.
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Obtuse Angles
An obtuse angle is an angle measuring between 90 and 180 degrees.
Example:
The following angles are all obtuse.
Right Angles
A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right angle are said to be perpendicular. Note that any two right angles are supplementary angles (a right angle is its own angle supplement).
Example:
The following angles are both right angles.
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Complementary Angles
Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the complementary angles is said to be the complement of the other.
Example:
These two angles (40° and 50°) are Complementary Angles, because they add up to 90°.
Notice that together they make a right angle.
But the angles don’t have to be together.
These two are complementary because 27° + 63° = 90°
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Supplementary Angles
Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. One of the supplementary angles is said to be the supplement of the other.
Two Angles are Supplementary if they add up to 180 degrees.
Example:
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But the angles don’t have to be together.
These two are supplementary because 60° + 120° = 180°
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If the two angles add to 180°, we say they “Supplement” each other.
Supplement comes from Latin supplere, to complete or “supply” what is needed.
Vertical Angles
For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Vertical angles have the same degree measurement. Angle BEC and angle AED are also vertical angles.
Vertical Angles are the angles opposite each other when two lines cross They are called “Vertical” because they share the same Vertex (or corner point)
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In this example, a° and b° are vertical angles.
The interesting thing here is that vertical angles are equal:
a° = b°
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Example: Find angles a°, b° and c° below:
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Because b° is vertically opposite 40°, it must also be 40° A full circle is 360°, so that leaves 360° - 2×40° = 280° Angles a° and c° are also vertical angles (and must be equal), which means they are 140° each. Answer: a = 140°, b = 40° and c = 140°.Note: They are also called Vertically Opposite Angles, which is just a more exact way of saying the same thing.
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Alternate Interior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles have the same degree measurement. Angle B and angle C are also alternate interior angles.
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When two lines are crossed by another line (which is called the Transversal), the pairs of angles on opposite sides of the transversal but inside the two lines are called Alternate Interior Angles.
In this example, these are Alternate Interior Angles:
c and f
d and e
(To help you remember: the angle pairs are on “Alternate” sides of the Transversal, and they are on the “Interior” of the two crossed lines)
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Alternate Exterior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior angles have the same degree measurement. Angle B and angle C are also alternate exterior angles.
When two lines are crossed by another line (which is called the Transversal), the angles in matching corners are called corresponding angles.
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Corresponding Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles. Corresponding angles have the same degree measurement. Angle B and angle D are also corresponding angles.
Corresponding Angles |
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| When two lines are crossed by another line (which is called the Transversal), the angles in matching corners are called corresponding angles. | |
Angle Bisector
An angle bisector is a ray that divides an angle into two equal angles.
Example:
The blue ray on the right is the angle bisector of the angle on the left.
The red ray on the right is the angle bisector of the angle on the left.
Perpendicular Lines
Two lines that meet at a right angle are perpendicular.